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Math Help - Multiplying Uniformly Continuous functions

  1. #1
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    Multiplying Uniformly Continuous functions

    If f and g are two uniformly continuous functions, I need to show that their product is also uniformly continuous. I started with the definition of uniform continuity but got nowhere...
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  2. #2
    Senior Member Tinyboss's Avatar
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    You're not going to get anywhere, either, unless you restrict the domain. Is x\mapsto x uniformly continuous on \mathbb{R}? What about x\mapsto x^2?
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    Sorry I'm not sure what you mean by that...
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  4. #4
    Senior Member Tinyboss's Avatar
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    Let f:\mathbb{R}\to\mathbb{R} be given by f(x)=x. Is f uniformly continuous? What about the product of f with f?
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    yea I haven't figured it out yet... it's probably simple but i'm having a lot of trouble with this one
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    f(x)=x is trivially uniformly continuous since \left| f(x)-f(y) \right|=\left| x-y \right|, so the product is not uniformly continuous, namely f(x)f(x)=x^2 is not uniformly continuous.
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  7. #7
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    lol but this is isn't making sense since I'm supposed to show the product IS uniformly continuous
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  8. #8
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    IT'S NOT TRUE, and i've provided a counterexample.
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  9. #9
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    lol you're right, now what if both functions were bounded?
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  10. #10
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    Quote Originally Posted by CrazyCat87 View Post
    yea I haven't figured it out yet... it's probably simple but i'm having a lot of trouble with this one
    I accept with information: is trivially uniformly continuous since
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